Covering functions by countably many functions
from some families
Zbigniew Grande
Institute of Mathematics, Kazimierz Wielki University, pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland (e-mail: grande@ukw.edu.pl)
Received March 22, 2013; revised July 24, 2013
Abstract. LetA be a nonempty family of functions from R to R. A function f : R → R is said to be strongly countably A-function if there is a sequence (fn) of functions from A such that Gr(f) ⊂ nGr(fn) (Gr(f) denotes the graph
of f ). IfA is the family of all continuous functions, the strongly countable A-functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81– 86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011].
In this article, we prove that the familiesA(R) of all strongly countably A-functions are closed with respect to some operations in dependence of analogous properties of the families A, and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly count-ably quasi-continuous functions.
MSC: primary 26A15; secondary 26A24, 26A21
Keywords: Lebesgue measurability, Baire property, first Baire class, approximate continuity, differentiability
1 Introduction
Let R be the set of all reals, and let A be a nonempty family of functions from R to R. A function
f : R → R is said to be strongly countably A-function if there is a sequence (fn) of functions from A such that Gr(f) ⊂ nGr(fn) (Gr(f) denotes the graph of f). If A is the family of all continuous func-tions, the strongly countable A-functions are called strongly countably continuous and were investigated in [4, 5, 7].
In this article, we prove that the families A(R) of all strongly countably A-functions are closed with respect to some operations in dependence of analogous properties of the families A. Moreover, we show some properties of strongly countably differentiable functions and strongly countably quasi-continuous func-tions. We recall that a function f : R → R is quasi-continuous at a point x ∈ R if, for each open inter-val I containing x and every real r > 0, there is an open interval J ⊂ I such that f(J) ⊂ (f(x) − r,
f(x) + r) (see [6, 8]).
2 The main results
2.1 The closedness ofA(A(A(R)R)R) with respect to some operations
We start from the obvious observation that if a familyA is contained in E, then A(R) ⊂ E(R).
Let F :R2 → R. Theorems 1 and 2 are generalizations of theorems from [4], and their proofs are similar to those from [4].
Theorem 1. Assume that the familyA is closed with respect to the operation F , i.e., for arbitrary functions
φ, ψ ∈ A, the function F (φ, ψ) ∈ A. Then, for arbitrary two functions f, g ∈ A(R), the function F (f, g) ∈
A(R).
Proof. There are sequences(fn) and (gn) of functions from A such that Gr(f) ⊂ n Gr(fn) and Gr(g) ⊂ n Gr(gn). For n= 1, 2, . . . , let An=x; fn(x) = f(x) and Bn=x; gn(x) = g(x). Evidently, n An= n Bn=R.
For n, k = 1, 2, . . . , put Dn,k = An∩ Bk. By our assumption the functions F(fn, gk) belong to A for
n, k = 1, 2, . . . . Moreover, for x ∈ Dn,k, we have F(f(x), g(x)) = F (fn(x), gk(x)). Since
R = n An= n Bn= n,k1 Dn,k, we obtain that GrF (f, g)⊂ n,k1 GrF (fn, gk) ,
and consequently, F(f, g) belongs to A(R).
From the above Theorem 1 we immediately get the following:
Corollary 1. Assume that the familyA is closed with respect to the addition (subtraction) [multiplication by
constant] {multiplication}. Then the familyA(R) has the same property.
For the investigation of the closedness of A(R) with respect to the division, we introduce the subfamily
A1(R) ⊂ A(R) such that f ∈ A1(R) if and only if there is a sequence of functions fn ∈ A, n = 1, 2, . . . , such thatGr(f) ⊂n1Gr(fn) and fn(R) ⊂ R \ {0} for n 1.
Observe that ifA is the family of all continuous functions or the family of all differentiable functions, then every function f ∈ A(R) with f(R) ⊂ R \ {0} belongs to A1(R).
Corollary 2. If, for arbitrary two functions φ, ψ ∈ A with ψ(R) ⊂ R \ {0}, the quotient φ/ψ belongs to A,
then, for arbitrary two functions f, g∈ A(R) with g ∈ A1(R), the quotient f/g belongs to A(R).
Proof. It suffices to repeat the proof of Theorem 1 for the operation F(x, y) = x/y defined onR×(R\{0}). The hypothesis that g∈ A1(R) implies that there is a sequence of functions gn∈ A such that gn(R) ⊂ R\{0} for n 1 and Gr(g) ⊂nGr(gn).
Similarly to Theorem 1, we obtain the following:
Corollary 3. Assume that the family A is closed with respect to the operation F1(x, y) = max(x, y) (F2(x, y) = min(x, y)). Then the family A(R) has the same property.
Theorem 2. Assume that the familyA is closed with respect to the superposition. Then the family A(R) has
the same property.
Proof. Fix two functions f, g∈ A(R). There are sequences (fn) and (gn) of functions from A such that Gr(f) ⊂ n Gr(fn) and Gr(g) ⊂ n Gr(gn). For n= 1, 2, . . . , let An=x; fn(x) = f(x) and Bn=x; gn(x) = g(x). Then n An= n Bn=R. For n, k= 1, 2, . . . , put Cn,k= (gn)−1(A k) ∩ Bn.
Observe that, for each point x ∈R, there is an index n1such that gn1(x) = g(x). Next, there is an index k1 with g(x) ∈ Ak1, that is,
fg(x)= fk1g(x)= fk1gn1(x). So, x ∈ Bn1 ∩ (gn1)−1(Ak1) = Cn1,k1 and R = n,k1 Cn,k. If x∈ Cn,k, then f(g(x)) = f(gn(x)) = fk(gn(x)). So, Grf(g)⊂ n,k1 Grfk(gn).
Since the compositions fk(gn), n, k 1, belong to A, the proof is complete.
The above results whenA is the family of all continuous functions were obtained in [4]. If A is the family of all constant functions, then, evidently, a function f ∈ A(R) (we will say that f is strongly countably constant) if and only if the image f(R) is countable. Now we will investigate some other cases.
2.2 Strongly countably differentiable functions
LetA be the family of all differentiable functions, which will be denoted by Δ. This family is closed with respect to the sums, differences, products, quotients (if the image of the denominator is contained inR \ {0}, and superpositions. It is not closed with respect to the operations max and min. Strongly countably Δ-functions will be called strongly countably differentiable. The characteristic Δ-functions of nonmeasurable (in the sense of Lebesgue) sets are strongly countably constant and, thus, also strongly countably differentiable. Evidently, such functions are nonmeasurable (and, thus, also discontinuous).
Theorem 3. Let f :R → R be a continuous function. Then f ∈ Δ(R) if and only if, for each nonempty closed
Proof. Necessity. Let f ∈ Δ(R) be a continuous function, and let H be a nonempty closed set. If H contains
isolated points, then the proof is evident. So, we can assume that H is a nonempty perfect set. There is a sequence(fn) of differentiable functions such that
Gr(f) ⊂ n
Gr(fn).
Since H is a complete space, there is an index n such that the set B = {x ∈ H; fn(x) = f(x)} is of the second category in H. But f and fnare continuous, so the set B is closed. So, there is an open interval I such that∅ = I ∩ H ⊂ B. To finish the proof of the necessity, it suffices to put gH = fn.
Sufficiency. For H =R, we find an open interval I0and a differentiable function f0such that f/I0 = f0/I0. Next, by transfinite induction we define a transfinite sequence of open intervals Iα, α < γ, with rational endpoints such that
Eα= Iα∩ R \ β<α Iβ = ∅,
and, for the restricted functions f/Eα, there are differentiable functions fα : R → R with fα/Eα = f/Eα. Since γ is a countable ordinal, the proof of the sufficiency is complete.
Remark 1. The referee observed that the same proof of Theorem 3 works if the hypothesis of the continuity
of the function f is replaced by f ∈ B1∗. Recall that f ∈ B1∗ if, for each nonempty closed set E, there is an open interval I with I∩ E = ∅, and the restriction f/(I ∩ E) is continuous (see [9]). Moreover, he observed a possibility of some generalization of Theorem 3 for some subfamiliesA ⊂ B∗1 instead of Δ.
Corollary 4. If f ∈ Δ(R) ∩ B1∗, then it is differentiable on an open dense set.
Corollary 5. If a continuous function f :R → R is nowhere differentiable, then it is not strongly countably
differentiable.
Remark 2. The referee informed me that the existence of a continuous function that cannot be covered by
countably many differentiable functions follows from theorem of Morayne (see Theorem 4.4 in [3]).
2.3 Strongly countably approximately continuous functions
We recall that the family of all measurable subsets A⊂R such that each point x ∈ A is a density point of A, i.e.,limh→0+m(A ∩ [x − h, x + h])/(2h) = 1, where m denotes the Lebesgue measure inR, is the so-called
density topology (compare [1]). Denote by Tdthe density topology and by Tethe natural topology inR. The continuity of functions from(R, Td) to (R, Te) is said to be the approximate continuity (see [1]). It is known that there are approximately continuous functions that are not strongly countably continuous (see [4]). In this section, we investigate strong countable approximate continuity. A function f :R → R is said to be strongly countably approximately continuous if there are approximately continuous functions fn : R → R such that Gr(f) ⊂nGr(fn). The approximate continuity of a function f :R → R implies the measurability (in the sense of Lebesgue) of its graph. So, the graphs of strongly countably approximately continuous functions are measurable.
Theorem 4. There are measurable functions f : R → R that are not strongly countably approximately
con-tinuous.
Proof. Let A ⊂ [0, 1] be a nowhere dense compact set of positive measure. The difference B of A and the union of all open intervals I with rational endpoints and m(A∩I) = 0 is a compact set such that m(B∩J) > 0 for each open interval J with B∩ J = ∅. Enumerate in a transfinite sequence (Cα)α<ωc (ωcdenotes the first ordinal of continuum cardinality) the family of all closed subsets of B×R that are of positive plane measure (in
the sense of Lebesgue). Next, by transfinite induction, we find a transfinite sequence of points(xα, yα) ∈ Cα,
α < ωc, such that xα = xβ for α= β, α, β < ωc. Let
g(xα) = yα for α < ωc and g(x) = 0 otherwise onR.
Then the graph of the restricted function g|B is not measurable in B × R. Since B is a nowhere dense closed set, there is (see [10, Thm. 13.1]) an autohomeomorphism h of[0, 1] such that h([0, 1]) = [0, 1] and
m(h−1(B)) = 0. Moreover, let
f(x) = gh(x) for x∈ h−1(B) and f(x) = 0 otherwise onR.
Since the values of the function f are equal0 for x ∈R \ h−1(B), it is measurable. We will prove that f is not strongly countably approximately continuous. Suppose that, contrary to our claim, there is a sequence of approximately continuous functions fn : R → R, n = 1, 2, . . . , such that Gr(f) ⊂ nGr(fn). Since fn,
n 1, are of the first Baire class, the functions φn(x) = fn(h−1(x)) for x ∈ R have the same property, and consequently, their graphs are of plane measure0. But Gr(g/B) ⊂ nGr(φn), so it is of plane measure 0. This contradicts the fact that the graphGr(g/B) is not measurable.
Remark 3. Observe that, for each countable family of Borel functions ψn : R → R, n 1, and for the
function f from the proof of Theorem 4, we haveGr(f) \nGr(ψn) = ∅.
Theorem 5. If a function f : R → R is of the first Baire class, then it is strongly countably approximately
continuous.
Proof. By Lusin’s theorem, for each positive integer n 2, there is a closed set An ⊂ [−n, n] such that the restriction f|An is continuous and m([−n, n] \ An) < 1/n. For each n 2, there is a continuous function (so also approximately continuous) fn : R → R such that fn/An = f/An. Let B = R \nAn. Observe that m(B) = 0. Indeed, suppose that, contrary to our claim, there is a positive real r with m(B) > r. Since m(B) = limn→∞m([−n, n] \knAk), there is an index k 2 with m([−n, n] \inAi) > r for
n k. But this contradicts the inequalities m([−n, n] \inAi) m([−n, n] \ An) < 1/n for n 2. So,
m(B) = 0. Since f is of the first Baire class and m(B) = 0, by Laczkovich and Petruska theorem from [7]
there is an approximately continuous function f1 : R → R such that f1/B = f/B. The evident inclusion Gr(f) ⊂nGr(fn) completes the proof.
The following problem is natural.
Problem 1. Does there exist functions f : R → R of the second Baire class that are not strongly countably
approximately continuous?
The referee informed me that the positive answer to this problem follows from Corollary 3.4 in [2] (see also [3], the foot of page 160), where the authors proved that, for each ordinal α < ω1, there is a function
f ∈ Bα+1 for which there is no countable partition{Xn; n ∈ ω} ofR such that f/Xn ∈ Bα(Xn). Clearly, such a function f cannot be covered by countably many functions from the class Bα. This result implies also Theorem 4 and Remark 3.
2.4 Strongly countably quasi-continuous functions
A function f : R → R is said to be strongly countably continuous if there is a sequence of quasi-continuous functions fn :R → R, n = 1, 2, . . . , such that Gr(f) ⊂
nGr(fn). Observe that the graphs of strongly countably quasi-continuous functions are of the first category on the planeR2 and there are strongly countably quasi-continuous functions that do not have the Baire property, for example, the characteristic func-tions of sets without the Baire property.
Theorem 6. If a function f :R → R has the Baire property, then it is strongly countably quasi-continuous.
Proof. Without loss of generality, we can assume that f is bounded; otherwise, we can consider the functions
ψn(x) = f(x) if |f(x)| n and ψn(x) = 0 otherwise on R and apply the observation that Gr(f) ⊂
nGr(ψn).
Since f has the Baire property, there is an Fσ-set A of the first category such that the restriction f|(R \ A) is continuous. Put f1(x) = f(x) for x ∈R \ A and f1(x) = lim supt∈R\A, t→xf(t) for x ∈ A. Observe that
f1is a quasi-continuous function.
Since A is an Fσ-set of the first category, there are nowhere dense closed sets An, n 2, such that
A =n2An. Now fix an integer n 2 and a component I of the set R \ An. If x is the right endpoint of the component I, then there are points ak(I) ∈ I such that ak(I) < ak+1(I) for k 1 and limk→∞ak(I) = x. Similarly, if y is the left endpoint of the component I, there are points bk(I) ∈ I such that a1(I) > b1(I) >
b2(I) > · · · and limk→∞bk(I) = y. For this component I, we define a continuous function φI : I →R such
that φI([a2k(I), a2k+1(I)]) = φI([b2k+1(I), b2k(I)]) = [−k, k] for k 1. Now let fn(x) = f(x) for x ∈ An and fn(x) = φI(x) on the components I of the complementR \ An. Evidently, the functions fn, n 2, are quasi-continuous andGr(f) ⊂n1Gr(fn).
3 Final problem
We have observed that all functions f :R → R of the first Baire class are strongly countably approximately continuous and strongly countably quasi-continuous.
Problem 2. Let f : R → R be a function of Baire 1 class. Do there exist a sequence of approximately
continuous and, simultaneously, quasi-continuous functions fn : R → R, n = 1, 2, . . . , such that Gr(f) ⊂
nGr(fn).
Acknowledgment. I wish to thank the referees for valuable remarks concerning the paper.
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